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Existence of non-trivial smooth solution of a diffusion equation with a dynamical boundary condition

Abstract: In this talk, I shall introduce some results on the existence of non-trivial smooth solution for a 2D diffusion equation, with a dynamical boundary. The reaction is restricted only on the boundary by a bistable term. The existence and uniqueness of H1 weak solution was given by Lax-Milgram theorem. And by modifying the proof of classical H2 regularity, it is possible to process the boundary term and the interior terms at the same time, as a result, we can upraise the regularity of the above H1 solution to C∞, and get the existence of nontrivial solutions by Bifurcation theory. There is also a branch of solutions being monotonic in some sense that can confirm rigorously the phenomenon of cell polarization.